Axiomatic Theories of Truth
Graham Leigh
University of Leeds
LC’08, 8th July 2008
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 1 / 15
Introduction Formalising Truth
Formalising Truth
Definition (Language of truth)
We work in LT the language of Peano Arithmetic augmented with anadditional predicate symbol T . Let PAT denote PA formulated in thelanguage LT .
The intuition is that T (x) denotes that x is (the Godel number of) a“true” LT sentence.Let p.q provide a Godel numbering of LT .
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 2 / 15
Introduction Formalising Truth
Formalising Truth
Definition (Language of truth)
We work in LT the language of Peano Arithmetic augmented with anadditional predicate symbol T . Let PAT denote PA formulated in thelanguage LT .
The intuition is that T (x) denotes that x is (the Godel number of) a“true” LT sentence.Let p.q provide a Godel numbering of LT .
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 2 / 15
Introduction Formalising Truth
Choices for truth
Of course, by Tarski’s Theorem the “ideal” axiom of truth, TpAq↔ A forall sentences A, is inconsistent with PAT . However, there are ways inwhich we can overcome this inconsistency.
1 Restrict the language so as to stop self-reference. For example allowTpAq↔ A for LPA.
2 Replace TpAq↔ A with weaker, consistent, axioms.
We will consider case 2.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 3 / 15
Introduction Formalising Truth
Choices for truth
Of course, by Tarski’s Theorem the “ideal” axiom of truth, TpAq↔ A forall sentences A, is inconsistent with PAT . However, there are ways inwhich we can overcome this inconsistency.
1 Restrict the language so as to stop self-reference. For example allowTpAq↔ A for LPA.
2 Replace TpAq↔ A with weaker, consistent, axioms.
We will consider case 2.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 3 / 15
Introduction Formalising Truth
Choices for truth
Of course, by Tarski’s Theorem the “ideal” axiom of truth, TpAq↔ A forall sentences A, is inconsistent with PAT . However, there are ways inwhich we can overcome this inconsistency.
1 Restrict the language so as to stop self-reference. For example allowTpAq↔ A for LPA.
2 Replace TpAq↔ A with weaker, consistent, axioms.
We will consider case 2.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 3 / 15
Introduction Formalising Truth
Choices for truth
Of course, by Tarski’s Theorem the “ideal” axiom of truth, TpAq↔ A forall sentences A, is inconsistent with PAT . However, there are ways inwhich we can overcome this inconsistency.
1 Restrict the language so as to stop self-reference. For example allowTpAq↔ A for LPA.
2 Replace TpAq↔ A with weaker, consistent, axioms.
We will consider case 2.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 3 / 15
Introduction A Base for truth
A Base for truth
Let BaseT be the theory comprising of PAT and the following axioms.
1 (TpA→ Bq ∧ TpAq)→ TpBq.
2 T (uclpBq) for all tautologies B.
3 TpAq if A is a true primitive recursive atomic sentence.
where ucl(A) denotes the (Godel number of the) universal closure of A.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 4 / 15
Introduction Axioms for truth.
Axioms of truth
Possible axioms, schema, and rules of inference we consider areA→ TpAq, ¬(TpAq ∧ Tp¬Aq), A/TpAq,TpAq→ A, TpAq ∨ Tp¬Aq, TpAq/A,TpAq→ TpTpAqq, ∀n TpAnq→ Tp∀x Axq, ¬A/¬TpAq,TpTpAqq→ TpAq, Tp∃x Axq→ ∃n TpAnq, ¬TpAq/¬A.
These axioms were considered by Harvey Freidman and Michael Sheard inAn axiomatic approach to self-referential truth [2].They classified the above axioms and rules into nine maximally consistentsets.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 5 / 15
Introduction Axioms for truth.
Axioms of truth
Possible axioms, schema, and rules of inference we consider areA→ TpAq, ¬(TpAq ∧ Tp¬Aq), A/TpAq,TpAq→ A, TpAq ∨ Tp¬Aq, TpAq/A,TpAq→ TpTpAqq, ∀n TpAnq→ Tp∀x Axq, ¬A/¬TpAq,TpTpAqq→ TpAq, Tp∃x Axq→ ∃n TpAnq, ¬TpAq/¬A.
These axioms were considered by Harvey Freidman and Michael Sheard inAn axiomatic approach to self-referential truth [2].They classified the above axioms and rules into nine maximally consistentsets.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 5 / 15
Ordinal Analyses Lower bounds
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
Definition
Let S1 be BaseT + U-Inf + T-Elim, andS2 be BaseT + U-Inf + T-Del + T-Intro + T-Elim.Denote by I (α) the formula ∀pAq TpTI(α, A)q.
Sheard proved (in [4]) that S1 ` ∀α. I (α)→ I (εα). Moreover he showed|S1| = ϕ20.We have shown S2 ` ∀α. I (α)→ I (ϕnα) for each n.
Proof. (Sketch).
From ` ∀α. I (α)→ I (ϕnα) we get ` ProgβI (ϕn′β) (with U-Inf). Thus,` ∀α. TIβ(α, I (ϕn′β))→ I (ϕn′α). Now by T-Intro, axioms of BaseT andT-Del we have ` ∀α. TpTIβ(α, I (ϕn′β))q→ I (ϕn′α). ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 6 / 15
Ordinal Analyses Lower bounds
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
Definition
Let S1 be BaseT + U-Inf + T-Elim, andS2 be BaseT + U-Inf + T-Del + T-Intro + T-Elim.Denote by I (α) the formula ∀pAq TpTI(α, A)q.
Sheard proved (in [4]) that S1 ` ∀α. I (α)→ I (εα). Moreover he showed|S1| = ϕ20.We have shown S2 ` ∀α. I (α)→ I (ϕnα) for each n.
Proof. (Sketch).
From ` ∀α. I (α)→ I (ϕnα) we get ` ProgβI (ϕn′β) (with U-Inf). Thus,` ∀α. TIβ(α, I (ϕn′β))→ I (ϕn′α). Now by T-Intro, axioms of BaseT andT-Del we have ` ∀α. TpTIβ(α, I (ϕn′β))q→ I (ϕn′α). ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 6 / 15
Ordinal Analyses Lower bounds
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
Definition
Let S1 be BaseT + U-Inf + T-Elim, andS2 be BaseT + U-Inf + T-Del + T-Intro + T-Elim.Denote by I (α) the formula ∀pAq TpTI(α, A)q.
Sheard proved (in [4]) that S1 ` ∀α. I (α)→ I (εα). Moreover he showed|S1| = ϕ20.We have shown S2 ` ∀α. I (α)→ I (ϕnα) for each n.
Proof. (Sketch).
From ` ∀α. I (α)→ I (ϕnα) we get ` ProgβI (ϕn′β) (with U-Inf). Thus,` ∀α. TIβ(α, I (ϕn′β))→ I (ϕn′α). Now by T-Intro, axioms of BaseT andT-Del we have ` ∀α. TpTIβ(α, I (ϕn′β))q→ I (ϕn′α). ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 6 / 15
Ordinal Analyses Lower bounds
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
Definition
Let S1 be BaseT + U-Inf + T-Elim, andS2 be BaseT + U-Inf + T-Del + T-Intro + T-Elim.Denote by I (α) the formula ∀pAq TpTI(α, A)q.
Sheard proved (in [4]) that S1 ` ∀α. I (α)→ I (εα). Moreover he showed|S1| = ϕ20.We have shown S2 ` ∀α. I (α)→ I (ϕnα) for each n.
Proof. (Sketch).
From ` ∀α. I (α)→ I (ϕnα) we get ` ProgβI (ϕn′β) (with U-Inf). Thus,` ∀α. TIβ(α, I (ϕn′β))→ I (ϕn′α). Now by T-Intro, axioms of BaseT andT-Del we have ` ∀α. TpTIβ(α, I (ϕn′β))q→ I (ϕn′α). ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 6 / 15
Ordinal Analyses Lower bounds
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
Definition
Let S1 be BaseT + U-Inf + T-Elim, andS2 be BaseT + U-Inf + T-Del + T-Intro + T-Elim.Denote by I (α) the formula ∀pAq TpTI(α, A)q.
Sheard proved (in [4]) that S1 ` ∀α. I (α)→ I (εα). Moreover he showed|S1| = ϕ20.We have shown S2 ` ∀α. I (α)→ I (ϕnα) for each n.
Proof. (Sketch).
From ` ∀α. I (α)→ I (ϕnα) we get ` ProgβI (ϕn′β) (with U-Inf). Thus,` ∀α. TIβ(α, I (ϕn′β))→ I (ϕn′α). Now by T-Intro, axioms of BaseT andT-Del we have ` ∀α. TpTIβ(α, I (ϕn′β))q→ I (ϕn′α). ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 6 / 15
Ordinal Analyses Lower bounds
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
Definition
Let S1 be BaseT + U-Inf + T-Elim, andS2 be BaseT + U-Inf + T-Del + T-Intro + T-Elim.Denote by I (α) the formula ∀pAq TpTI(α, A)q.
Sheard proved (in [4]) that S1 ` ∀α. I (α)→ I (εα). Moreover he showed|S1| = ϕ20.We have shown S2 ` ∀α. I (α)→ I (ϕnα) for each n.
Proof. (Sketch).
From ` ∀α. I (α)→ I (ϕnα) we get ` ProgβI (ϕn′β) (with U-Inf). Thus,` ∀α. TIβ(α, I (ϕn′β))→ I (ϕn′α). Now by T-Intro, axioms of BaseT andT-Del we have ` ∀α. TpTIβ(α, I (ϕn′β))q→ I (ϕn′α). ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 6 / 15
Ordinal Analyses Infinitary Theories
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
It is more interesting to find an upper bound for S2. For this we need totake a detour into infinitary logic.
Definition (Inductive Definition of S∞2 )
Define S∞2α,n
kΓ by (Ax.1), (∧), (∨i ), (ω), (∃) and
(Cut). Ifα, n
kΓ,A,
δ, n
kΓ,¬A and |A| < k then
β, n
kΓ,
(Ax.2.).α, n
kΓ,¬T (A),T (A) ,
(Ax.3.).α, n
kΓ,¬T (A) if A is not an LT -sentence,
(T-Intro). Ifα, n
kA and n < m then
β,m
kΓ,T (A),
(T-Imp). Ifα, n
kΓ,T (A),
δ, n
kΓ,T (A→ B)) then
β, n
kΓ,T (B),
(T-Del). Ifα, n
kΓ,TpT (A)q then
β, n
kΓ,T (A),
(T-U-Inf). Ifα, n
kΓ,TpAmq for all m,
β, n
kΓ,T (∀x Ax),
if α, δ < β.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 7 / 15
Ordinal Analyses Infinitary Theories
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
It is more interesting to find an upper bound for S2. For this we need totake a detour into infinitary logic.
Definition (Inductive Definition of S∞2 )
Define S∞2α,n
kΓ by (Ax.1), (∧), (∨i ), (ω), (∃) and
(Cut). Ifα, n
kΓ,A,
δ, n
kΓ,¬A and |A| < k then
β, n
kΓ,
(Ax.2.).α, n
kΓ,¬T (A),T (A) ,
(Ax.3.).α, n
kΓ,¬T (A) if A is not an LT -sentence,
(T-Intro). Ifα, n
kA and n < m then
β,m
kΓ,T (A),
(T-Imp). Ifα, n
kΓ,T (A),
δ, n
kΓ,T (A→ B)) then
β, n
kΓ,T (B),
(T-Del). Ifα, n
kΓ,TpT (A)q then
β, n
kΓ,T (A),
(T-U-Inf). Ifα, n
kΓ,TpAmq for all m,
β, n
kΓ,T (∀x Ax),
if α, δ < β.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 7 / 15
Ordinal Analyses Infinitary Theories
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
It is more interesting to find an upper bound for S2. For this we need totake a detour into infinitary logic.
Definition (Inductive Definition of S∞2 )
Define S∞2α,n
kΓ by (Ax.1), (∧), (∨i ), (ω), (∃) and
(Cut). Ifα, n
kΓ,A,
δ, n
kΓ,¬A and |A| < k then
β, n
kΓ,
(Ax.2.).α, n
kΓ,¬T (A),T (A) ,
(Ax.3.).α, n
kΓ,¬T (A) if A is not an LT -sentence,
(T-Intro). Ifα, n
kA and n < m then
β,m
kΓ,T (A),
(T-Imp). Ifα, n
kΓ,T (A),
δ, n
kΓ,T (A→ B)) then
β, n
kΓ,T (B),
(T-Del). Ifα, n
kΓ,TpT (A)q then
β, n
kΓ,T (A),
(T-U-Inf). Ifα, n
kΓ,TpAmq for all m,
β, n
kΓ,T (∀x Ax),
if α, δ < β.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 7 / 15
Ordinal Analyses Infinitary Theories
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
It is more interesting to find an upper bound for S2. For this we need totake a detour into infinitary logic.
Definition (Inductive Definition of S∞2 )
Define S∞2α,n
kΓ by (Ax.1), (∧), (∨i ), (ω), (∃) and
(Cut). Ifα, n
kΓ,A,
δ, n
kΓ,¬A and |A| < k then
β, n
kΓ,
(Ax.2.).α, n
kΓ,¬T (A),T (A) ,
(Ax.3.).α, n
kΓ,¬T (A) if A is not an LT -sentence,
(T-Intro). Ifα, n
kA and n < m then
β,m
kΓ,T (A),
(T-Imp). Ifα, n
kΓ,T (A),
δ, n
kΓ,T (A→ B)) then
β, n
kΓ,T (B),
(T-Del). Ifα, n
kΓ,TpT (A)q then
β, n
kΓ,T (A),
(T-U-Inf). Ifα, n
kΓ,TpAmq for all m,
β, n
kΓ,T (∀x Ax),
if α, δ < β.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 7 / 15
Ordinal Analyses Infinitary Theories
U-Inf: ∀n TpAnq→ Tp∀x Axq T-Elim: TpAq/AT-Del: TpTpAqq→ TpAq T-Intro: A/TpAq
It is more interesting to find an upper bound for S2. For this we need totake a detour into infinitary logic.
Definition (Inductive Definition of S∞2 )
Define S∞2α,n
kΓ by (Ax.1), (∧), (∨i ), (ω), (∃) and
(Cut). Ifα, n
kΓ,A,
δ, n
kΓ,¬A and |A| < k then
β, n
kΓ,
(Ax.2.).α, n
kΓ,¬T (A),T (A) ,
(Ax.3.).α, n
kΓ,¬T (A) if A is not an LT -sentence,
(T-Intro). Ifα, n
kA and n < m then
β,m
kΓ,T (A),
(T-Imp). Ifα, n
kΓ,T (A),
δ, n
kΓ,T (A→ B)) then
β, n
kΓ,T (B),
(T-Del). Ifα, n
kΓ,TpT (A)q then
β, n
kΓ,T (A),
(T-U-Inf). Ifα, n
kΓ,TpAmq for all m,
β, n
kΓ,T (∀x Ax),
if α, δ < β.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 7 / 15
Ordinal Analyses Infinitary Theories
Definition
The rank of A, |A|, is defined as follows.
|A| = 0 if A is an arithmetical literal or T (s) for some term s.
|A ∧ B| = |A ∨ B| = |∀x A| = |∃x A| = |A|+ 1.
Theorem
Cut EliminationS∞2
α, n
k+1Γ implies S∞2
ωα, n
kΓ .
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 8 / 15
Ordinal Analyses Upper bounds
For each n > 0 and α define
Mn, α =⟨N, {pBq : S∞2
α0,m
0B for some m < n and α0 < α}
⟩and define M0, α = 〈N, ∅〉.
Lemma
For each n define fn(α) = ϕn(ϕ1α). Then for every n < ω we have
1 Ifα, n
0Γ then Mn, fn(α) |= Γ.
2 Ifα, n
0TpAq then
fn(α), p
0A for some p < n.
3 Ifα, n
kΓ then
ϕ1α, n
0Γ.
Corollary
If α < ϕω0 thenα,n
0TpAq implies
β,n
0A for some β < ϕω0.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 9 / 15
Ordinal Analyses Upper bounds
For each n > 0 and α define
Mn, α =⟨N, {pBq : S∞2
α0,m
0B for some m < n and α0 < α}
⟩and define M0, α = 〈N, ∅〉.
Lemma
For each n define fn(α) = ϕn(ϕ1α). Then for every n < ω we have
1 Ifα, n
0Γ then Mn, fn(α) |= Γ.
2 Ifα, n
0TpAq then
fn(α), p
0A for some p < n.
3 Ifα, n
kΓ then
ϕ1α, n
0Γ.
Corollary
If α < ϕω0 thenα,n
0TpAq implies
β,n
0A for some β < ϕω0.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 9 / 15
Ordinal Analyses Upper bounds
For each n > 0 and α define
Mn, α =⟨N, {pBq : S∞2
α0,m
0B for some m < n and α0 < α}
⟩and define M0, α = 〈N, ∅〉.
Lemma
For each n define fn(α) = ϕn(ϕ1α). Then for every n < ω we have
1 Ifα, n
0Γ then Mn, fn(α) |= Γ.
2 Ifα, n
0TpAq then
fn(α), p
0A for some p < n.
3 Ifα, n
kΓ then
ϕ1α, n
0Γ.
Corollary
If α < ϕω0 thenα,n
0TpAq implies
β,n
0A for some β < ϕω0.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 9 / 15
Ordinal Analyses Upper bounds
Thus, we have
Lemma
If S2 ` A then S∞2α,n
0A for some α < ϕω0.
and
Theorem
Let A be an arithmetical sentence, then S2 ` A impliesPA + TI(<ϕω0) ` A.
Hence
Corollary (T-Elimination for S∞2 )
|S2| = ϕω0.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 10 / 15
Ordinal Analyses Upper bounds
Thus, we have
Lemma
If S2 ` A then S∞2α,n
0A for some α < ϕω0.
and
Theorem
Let A be an arithmetical sentence, then S2 ` A impliesPA + TI(<ϕω0) ` A.
Hence
Corollary (T-Elimination for S∞2 )
|S2| = ϕω0.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 10 / 15
The theory E
U-Inf: ∀n TpAnq→ Tp∀x Axq, T-Elim: TpAq/A, T-Cons:T-Del: TpTpAqq→ TpAq, T-Intro: A/TpAq ¬(TpAq ∧ Tp¬Aq)
The bounds for S2 were fairly easy to establish. However, this is not thecase for all nine of the theories we considered.
For example, E is given by
BaseT + T-Del + U-Inf + T-Cons + T-Intro + T-Elim.
The upper bound E is not so clear because we no longer haveCut-Elimination in the corresponding infinitary system. However, we canembed E in a small extension of ID∗1.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 11 / 15
The theory E
U-Inf: ∀n TpAnq→ Tp∀x Axq, T-Elim: TpAq/A, T-Cons:T-Del: TpTpAqq→ TpAq, T-Intro: A/TpAq ¬(TpAq ∧ Tp¬Aq)
The bounds for S2 were fairly easy to establish. However, this is not thecase for all nine of the theories we considered.
For example, E is given by
BaseT + T-Del + U-Inf + T-Cons + T-Intro + T-Elim.
The upper bound E is not so clear because we no longer haveCut-Elimination in the corresponding infinitary system. However, we canembed E in a small extension of ID∗1.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 11 / 15
The theory E ID∗+1
ID∗1 is the theory extending PRA in which for each arithmetical formulaA ∈ L+
P with only one free variable the language is augmented by anadditional predicate symbol IA and we have the axioms
∀u. A(u, IA)→ IA(u), (Ax.IA.1)
∀u[A(u,F )→ F (u)]→ ∀u[IA(u)→ F (u)], (Ax.IA.2)
for each formula F containing only positive occurrences of predicates IBfor B ∈ L+
P and induction for formulae where fixed-point predicates occurpositively.We define ID∗+1 to be ID∗1 with, as an additional axiom,
∀u[A(u,F )→ F (u)]→ ∀u[IA(u)→ F (u)], (Ax.IA.3)
if A ∈ L+P is Σ2 and F is any formula which is Σ1 or Π1 in IA and ¬IA
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 12 / 15
The theory E ID∗+1
ID∗1 is the theory extending PRA in which for each arithmetical formulaA ∈ L+
P with only one free variable the language is augmented by anadditional predicate symbol IA and we have the axioms
∀u. A(u, IA)→ IA(u), (Ax.IA.1)
∀u[A(u,F )→ F (u)]→ ∀u[IA(u)→ F (u)], (Ax.IA.2)
for each formula F containing only positive occurrences of predicates IBfor B ∈ L+
P and induction for formulae where fixed-point predicates occurpositively.We define ID∗+1 to be ID∗1 with, as an additional axiom,
∀u[A(u,F )→ F (u)]→ ∀u[IA(u)→ F (u)], (Ax.IA.3)
if A ∈ L+P is Σ2 and F is any formula which is Σ1 or Π1 in IA and ¬IA
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 12 / 15
The theory E E and inductive definitions
Theorem
There are formulae An(u,P+) such that E ` C implies there is an n suchthat ID∗+1 ` In(pC ∗q).
Theorem
Every arithmetical consequence of E is a theorem of ID∗+1 .
Proof.
Let A be a model for the first-order part of ID∗+1 . Using A we may thenbuild a hierarchy of LT -structures
M0 = 〈A, ∅〉 ;
Mn+1 = 〈A, In〉 .
with the property that Mn |= In. Now, if E ` A then Mn |= A for some n.Hence A |= A. But A was arbitrary. ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 13 / 15
The theory E E and inductive definitions
Theorem
There are formulae An(u,P+) such that E ` C implies there is an n suchthat ID∗+1 ` In(pC ∗q).
Theorem
Every arithmetical consequence of E is a theorem of ID∗+1 .
Proof.
Let A be a model for the first-order part of ID∗+1 . Using A we may thenbuild a hierarchy of LT -structures
M0 = 〈A, ∅〉 ;
Mn+1 = 〈A, In〉 .
with the property that Mn |= In. Now, if E ` A then Mn |= A for some n.Hence A |= A. But A was arbitrary. ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 13 / 15
The theory E E and inductive definitions
Theorem
There are formulae An(u,P+) such that E ` C implies there is an n suchthat ID∗+1 ` In(pC ∗q).
Theorem
Every arithmetical consequence of E is a theorem of ID∗+1 .
Proof.
Let A be a model for the first-order part of ID∗+1 . Using A we may thenbuild a hierarchy of LT -structures
M0 = 〈A, ∅〉 ;
Mn+1 = 〈A, In〉 .
with the property that Mn |= In. Now, if E ` A then Mn |= A for some n.Hence A |= A. But A was arbitrary. ut
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 13 / 15
The theory E E and inductive definitions
And so,
Theorem
ϕω0 = |ID∗1| ≤ |E| ≤ |ID+1 |.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 14 / 15
References
References
Thank you.
A. Cantini.A theory of formal truth arithmetically equivalent to ID1.Journal of Symbolic Logic, 55(1):244–259, March 1990.
H. Friedman and M. Sheard.An axiomatic approach to self-referential truth.Annals of Pure and Applied Logic, 33:1–21, 1987.
V. Halbach.A system of complete and consistent truth.Notre Dame Journal of Formal Logic, 35(3), 1994.
M. Sheard.Weak and strong theories of truth.Studia Logica, 68:89–101, 2001.
Graham Leigh (University of Leeds) Axiomatic Theories of Truth LC’08, 8th July 2008 15 / 15